Answer
$(1 - i)^3$
= $- 2 - 2i$
Work Step by Step
$1 - i$ is at $315^\circ$ with absolute value $\sqrt{(1)^2 + (-1)^2} = \sqrt{2}$, the equivalent trigonometric form of $ 1 - i$ is $\sqrt{2}cis315^\circ$.
$(1 - i)^3$
= $(\sqrt{2}cis315^\circ)^3$
= $(\sqrt{2})^3cis3\cdot 315^\circ$ (De Moivre’s theorem)
= $(\sqrt{2})^3cis945^\circ$
= $(\sqrt{2})^3cis225^\circ$ ($945^\circ$ and $225^\circ$ are coterminal)
= $(\sqrt{2})^3 (- \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}i)$
= $- 2 - 2i$
